January  2020, 16(1): 309-324. doi: 10.3934/jimo.2018153

On the M-eigenvalue estimation of fourth-order partially symmetric tensors

1. 

School of Mathematics and Information Science, Weifang University, Weifang Shandong, 261061, China

2. 

School of Management Science, Qufu Normal University, Rizhao Shandong, 276826, China

* Corresponding author: Haitao Che

Received  January 2018 Revised  May 2018 Published  September 2018

Fund Project: This project is supported by the Natural Science Foundation of China (11401438, 11671228, 11601261, 11571120), Shandong Provincial Natural Science Foundation (ZR2016AQ12), Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J14LI52), and China Postdoctoral Science Foundation (Grant No. 2017M622163, 2018T110669)

In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.

Citation: Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
References:
[1]

K. ChangK. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[2]

H. ChenZ. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[4]

H. ChenZ. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar

[6]

H. ChenL. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[7]

S. Chirit$\check{a}$A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. Google Scholar

[10]

D. HanH. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar

[11]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.  Google Scholar

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.  Google Scholar

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. QiH. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar

[18]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar

[21]

Y. WangL. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[22]

X. WangH. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[23]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar

[24]

Y. WangK. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[25]

G. WangG. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

show all references

References:
[1]

K. ChangK. Pearson and T. Zhang, Some variational principles for Z-eigenvalues of nonnegative tensors, Linear Algebra Appl., 438 (2013), 4166-4182.  doi: 10.1016/j.laa.2013.02.013.  Google Scholar

[2]

H. ChenZ. Huang and L. Qi, Copositivity detection of tensors: Theory and algorithm, J. Optimiz. Theory Appl., 174 (2017), 746-761.  doi: 10.1007/s10957-017-1131-2.  Google Scholar

[3]

H. Chen, Y. Chen, G. Li and L. Qi, A semi-definite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl., 25 (2018), e2125, 16pp. doi: 10.1002/nla.2125.  Google Scholar

[4]

H. ChenZ. Huang and L. Qi, Copositive tensor detection and its applications in physics and hypergraphs, Comput. Optim. Appl., 69 (2018), 133-158.  doi: 10.1007/s10589-017-9938-1.  Google Scholar

[5]

H. Chen and Y. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China, 12 (2017), 1289-1302.  doi: 10.1007/s11464-017-0645-0.  Google Scholar

[6]

H. ChenL. Qi and Y. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China, 13 (2018), 255-276.  doi: 10.1007/s11464-018-0681-4.  Google Scholar

[7]

S. Chirit$\check{a}$A. Danescu and M. Ciarletta, On the srtong ellipticity of the anisotropic linearly elastic materials, J. Elasticity, 87 (2007), 1-27.  doi: 10.1007/s10659-006-9096-7.  Google Scholar

[8]

B. Dacorogna, Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension, Discrete Cont. Dyn-B, 1 (2001), 257-263.  doi: 10.3934/dcdsb.2001.1.257.  Google Scholar

[9]

W. Ding, J. Liu, L. Qi and H. Yan, Elasticity M-tensors and the strong ellipticity condition, preprint, arXiv: 1705.09911. Google Scholar

[10]

D. HanH. Dai and L. Qi, Conditions for strong ellipticity of anisotropic elastic materials, J. Elasticity, 97 (2009), 1-13.  doi: 10.1007/s10659-009-9205-5.  Google Scholar

[11]

J. He and T. Huang, Upper bound for the largest Z-eigenvalue of positive tensors, Appl. Math. Lett., 38 (2014), 110-114.  doi: 10.1016/j.aml.2014.07.012.  Google Scholar

[12]

Z. Huang and L. Qi, Positive definiteness of paired symmetric tensors and elasticity tensors, J. Comput. Appl. Math., 338 (2018), 22-43.  doi: 10.1016/j.cam.2018.01.025.  Google Scholar

[13]

J. K. Knowles and E. Sternberg, On the ellipticity of the equations of non-linear elastostatics for a special material, J. Elasticity, 5 (1975), 341-361.  doi: 10.1007/BF00126996.  Google Scholar

[14]

J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Arch. Rational Mech. Anal., 63 (1977), 321-336.  doi: 10.1007/BF00279991.  Google Scholar

[15]

E. Kofidis and P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors, SIAM J. Matrix Anal. Appl., 23 (2002), 863-884.  doi: 10.1137/S0895479801387413.  Google Scholar

[16]

C. Padovani, Strong ellipticity of transversely isotropic elasticity tensors, Meccanica, 37 (2002), 515-525.  doi: 10.1023/A:1020946506754.  Google Scholar

[17]

L. QiH. Dai and D. Han, Conditions for strong ellipticity and M-eigenvalues, Front. Math. China, 4 (2009), 349-364.  doi: 10.1007/s11464-009-0016-6.  Google Scholar

[18]

L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 40 (2005), 1302-1324.  doi: 10.1016/j.jsc.2005.05.007.  Google Scholar

[19]

Y. Song and L. Qi, Spectral properties of positively homogeneous operators induced by higher order tensors, SIAM J. Matrix Anal. Appl., 34 (2013), 1581-1595.  doi: 10.1137/130909135.  Google Scholar

[20]

J. R. Walton and J. P. Wilber, Sufficient conditions for strong ellipticity for a class of anisotropic materials, Int. J. Nonlin. Mech., 38 (2003), 441-455.  doi: 10.1016/S0020-7462(01)00066-X.  Google Scholar

[21]

Y. WangL. Caccetta and G. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra and Appl., 22 (2015), 1059-1076.  doi: 10.1002/nla.1996.  Google Scholar

[22]

X. WangH. Chen and Y. Wang, Solution structures of tensor complementarity problem, Front. Math. China, 13 (2018), 935-945.  doi: 10.1007/s11464-018-0675-2.  Google Scholar

[23]

Y. WangL. Qi and X. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl., 16 (2009), 589-601.  doi: 10.1002/nla.633.  Google Scholar

[24]

Y. WangK. Zhang and H. Sun, Criteria for strong H-tensors, Front. Mathe. China, 11 (2016), 577-592.  doi: 10.1007/s11464-016-0525-z.  Google Scholar

[25]

G. WangG. Zhou and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Cont. Dyn-B, 22 (2017), 187-198.  doi: 10.3934/dcdsb.2017009.  Google Scholar

[26]

T. Zhang and G. Golub, Rank-1 approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), 534-550.  doi: 10.1137/S0895479899352045.  Google Scholar

[27]

K. Zhang and Y. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms, J. Comput. Appl. Math., 305 (2016), 1-10.  doi: 10.1016/j.cam.2016.03.025.  Google Scholar

[28]

G. Zhou, G. Wang, L. Qi and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl., 25 (2018), e2134, 10pp. doi: 10.1002/nla.2134.  Google Scholar

Figure 1.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$, $ \mathcal{M}(\mathcal{C})$ and $ \mathcal{N}(\mathcal{C})$
Figure 2.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
Figure 3.  The comparisons of $ \Gamma(\mathcal{C})$, $ \mathcal{L}(\mathcal{C})$ and $ \mathcal{M}(\mathcal{C})$
[1]

Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 187-198. doi: 10.3934/dcdsb.2017009

[2]

Jun He, Guangjun Xu, Yanmin Liu. Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2019092

[3]

Yining Gu, Wei Wu. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 775-789. doi: 10.3934/jimo.2018070

[4]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042

[5]

Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019039

[6]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019129

[7]

Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial & Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975

[8]

Zhen Wang, Wei Wu. Bounds for the greatest eigenvalue of positive tensors. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1031-1039. doi: 10.3934/jimo.2014.10.1031

[9]

Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019069

[10]

Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial & Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054

[11]

Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263

[12]

Lixing Han. An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 583-599. doi: 10.3934/naco.2013.3.583

[13]

Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033

[14]

Gang Meng. The optimal upper bound for the first eigenvalue of the fourth order equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6369-6382. doi: 10.3934/dcds.2017276

[15]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[16]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729

[17]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019097

[18]

Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75

[19]

Wenjuan Zhai, Bingzhen Chen. A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 71-84. doi: 10.3934/naco.2019006

[20]

Zhongming Chen, Liqun Qi. Circulant tensors with applications to spectral hypergraph theory and stochastic process. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1227-1247. doi: 10.3934/jimo.2016.12.1227

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (86)
  • HTML views (781)
  • Cited by (0)

Other articles
by authors

[Back to Top]